Remainder Theorem

Hey, i have two questions..

Determine the remainder when (3x^5-5x^2+4x+1) is divided by (x-1)(x+2).
*Should i factor out (x-1)(x+2) into x^2+x-2 and then divide?

When x+2 is divided by f(x), the remainder is 3. Determine the remainder when x+2 is divided into each of the following:
a) f(x)+1
b) 2f(x)-7
...then there are about ten more , but im sure just as soon as i understand the first two the others will come easier.
For these ones i am totally lost on what to do...

Thanks so much for any help and sorry i dont have more done....

bandaid-bandet said:

Determine the remainder when (3x^5-5x^2+4x+1) is divided by (x-1)(x+2).

Click to expand...

I'm not sure how you're supposed to use the Remainder Theorem for this...? I would divide by x - 1 and then x + 2 (or the other way 'round), or else divide by their product, x² + x - 2.

bandaid-bandet said:

When x+2 is divided by f(x), the remainder is 3. Determine the remainder when x+2 is divided into each of the following:
a) f(x) + 1

Click to expand...

Because of the remainder, you have, for some (unspecified) polynomial "p(x)":

. . . . .f(x) = (x + 2)[p(x)] + 3

(This is like when you divide 13 by 4: you get 3, with a remainder of 1. This means that 13 = (3)[4] + 1. I'm doing the same thing in the indented bit above.)

So what happens when you add "1" to both sides? What is the remainder?

bandaid-bandet said:

b) 2f(x) - 7

Click to expand...

Do the same sort of thing:

. . . . .f(x) = (x + 2)[p(x)] + 3

. . . . .2f(x) = 2[(x + 2)[p(x)] + 3]
. . . . . . . .. .= (x + 2)[2p(x)] + ??

What's the remainder now?

They should all work similarly. If you're stuck or not sure, please reply showing what you've done. Thank you.

Eliz.